elevation, pitch and travel axis stabilization of 3dof

International Journal of Electrical and Computer Engineering (IJECE)

Vol. 10, No. 2, April 2020, pp. 1868~1884

ISSN: 2088-8708, DOI: 10.11591/ijece.v10i2.pp1868-1884 1868

Journal homepage: http://ijece.iaescore.com/index.php/IJECE

Elevation, pitch and travel axis stabilization of 3DOF helicopter

with hybrid control system by GA-LQR based PID controller

Ibrahim K. Mohammed, Abdulla I. Abdulla Department of Systems and Control Engineering, College of Electronics Engineering, Ninevah University, Iraq

Article Info ABSTRACT

Article history:

Received Apr 29, 2019

Revised Aug 28, 2019

Accepted Oct 9, 2019

This research work introduces an efficient hybrid control methodology

through combining the traditional proportional-integral-derivative (PID)

controller and linear quadratic regulator (LQR) optimal controlher.

The proposed hybrid control approach is adopted to design three degree of

freedom (3DOF) stabilizing system for helicopter. The gain parameters of

the classic PID controller are determined using the elements of the LQR

feedback gain matrix. The dynamic behaviour of the LQR based PID

controller, is modeled in state space form to enable utlizing state feedback

controller technique. The performance of the proposed LQR based LQR

controller is improved by using Genetic Algorithm optimization method

which are adopted to obtain optimum values for LQR controller gain

parameters. The LQR-PID hybrid controller is simulated using Matlab

environment and its performance is evaluated based on rise time, settling

time, overshoot and steady state error parameters to validate the proposed

3DOF helicopter balancing system. Based on GA tuning approach,

the simulation results suggest that the hybrid LQR-PID controller can be

effectively employed to stabilize the 3DOF helicopter system.

Keywords:

3DOF helicopter system

Genetic algorithm

Hybrid control

LQR controller

PID controller

Copyright © 2020 Institute of Advanced Engineering and Science.

All rights reserved.

Corresponding Author:

Ibrahim K. Mohammed,

Department of Systems and Control Engineering,

College of Electronics Engineeing,

Ninevah University, Mosul, Iraq

Email: [email protected]

1. INTRODUCTION

The A 3DOF helicopter model is a multi-input, multi-output system (MIMO) with strongly

nonlinear characteristics, used as experimental platform for verification of various control algorithms.

The helicopter is a complex mechanical scheme and has unstable dynamics which make using traditional

control methods to stabilize it is a big challenge. In control procedure of 3DOF helicopter system

the nonlinear dynamics is linearised about preselected operating conditions. Based on linearised system

model a classical proportional integral derivative PID controller technique is widely used due to its simplicity

and easy to realize [1]. However, using conventional PID technique is insufficient to guide the helicopter

through the demand trajectory as the regulation of the controller parameters is hard task even for

an experienced control engineers [1, 2]. In another control method, an optimal tracking approach uilizing

LQR controller to stabilize the helicopter system was proposed by [3]. In the proposed system,

a Genetic Algorithm (GA) is employed to optimize a classic LQR controller for position control of DC

motor. By using LQR controller a significant improvement in guidance of helicopter system is carried out.

Further investigations in control strategies for helicopter system are still the goal of many researchers. A new

control approach based on hybrid controller technique was proposed by many researchers to control

the 3DOF helicopter system. Fan and Joo [4] proposed a hybrid control method using fuzzy and PID

controller to balance the 3DOF helicopter system at desired equilibrium position. In this study, an efficient

Int J Elec & Comp Eng ISSN: 2088-8708

Elevation, pitch and travel axis stabilization of 3DOF helicopter with… (Ibrahim K. Mohammed)

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control method has been adopted to propose a hybrid control strategy using both conventional and intelligent

control algorithms. This hybrid control approach uses PID controller as a conventional control, while

the liner quadratic regulator (LQR) controller based on GA used as an intelligent control. The LQR optimal

controller is recommended to implement the intelligent system due to its high precision in movement

applications [5]. In this state feedback controller technique a tradeoff between the plant characteristics and

the input effort of the system can be achieved if the controller gain parameters are tuned propperly [6-8].

The LQR controller design can be improved by employing soft computing and optimization techniques,

which are adopted to achieve tuning proecss of the controller gain parameters, such as Differential Evolution

(DE) [9], Particle Swarm Inspired Evolutionary Algorithm (PS-EA) [10], Particle Swarm Optimisation

(PSO) [11], Genetic Algorithm (GA) [12], memetic algorithm (MA) [13], Imperialist Competitive Algorithm

(ICA) [14], Ant Colony Optimization (ACO) [15], Artificial Bee Colony (ABC) [16], and Artificial Immune

Systems (AIS) [17]. In order to validate the proposed helicopter model, the hybrid control system is

simulated by using the Matlab environment, and its balancing performance is then evaluated based on

stability criteria parameters which include rise and settling time, overshoot and control input. The purpose of

the proposed hybrid GA-LQR based PID controller is to design an efficient control system utilized to

stabilize a prototype 3DOF helicopter system at desired roll and pitch positions as well as angular travel

speed. During the current decade there has been a considerable interest by many control researchers in using

of hybrid control approach in various engineering applications. Vendoh and Jovitha [18] presented a hybrid

control system using LQR based PID controller for stabilization and trajectory tracking of magnetic levitation

system. Arbab et al. [19] proposed a hybrid control strategy using fuzzy based LQR controller for 3-DOF

helicopter system. Choudhary [1] design another hybrid control scheme using LQR based PID controller for

3-DOF helicopter system, however, the introduced hybrid control systems are not efficient as the parameters

of the LQR feedback gain matrix are obtained by applying trial and error procedure. The paper presents

a simple method for the approximation of PID controller gain parameters from feedback gain matrix of

the controller. The LQR controller gain is obtained from the weighting matrices Q and R which their

elements values are tuned effectively by the GA optimization method. The rest of the paper is organised as follows. Section 2 presents configuration and dynamics

modeling of the helicopter system. In section 3, controller techniques of helicopter hybrid control system are

introduced. Tuning method is described in section 4. Section 5 introduces helicopter control system design is

presented. In section 6 simulation results of the hybrid control system are introduced and followed by

conclusions and prospective research in section 7.

2. STRUCTURE AND DYNAMICS MODELING OF HELICOPTER

2.1. Helicopter stucture

The conceptual platform of 3-DOF helicopter system is shown in Figure 1. It consists of an arm

mounted on a base. The main body of the helicopter composed of propellers driven by two motors mounted

are the either ends of an short balance bar. The whole helicopter body is fixed on one end of the arm and

a balance block installed at the other end. The balance arm can rotate about travel axis as well as slope on an

elevation axis. The body of helicopter is free to roll about the pitch axis. The system is provided by encoders

mounted on these axis used to measure the travel motion of the arm and its elevation and pitch angle.

The propellers with motors can generate an elevation mechanical force proportional to the voltage power

supplied to the motors. This force can cause the helicopter body to lift off the ground. It is worth considering

that the purpose of using balance block is to reduce the voltage power supplied to the propellers motors.

2.2. Helicopter dynamics and mathematical modeling

In this study, the nonlinear dynamics of 3DOF helicopter system is modeled mathematically based

on developing the model of the system behavior for elevation, pitch and travel axis. The definition of

the symbols and nomecalture of the proposed helicopter system is included in Table 1.

2.2.1. Elevation axis model

The free body diagram of 3DOF helicopter system based on elevation axis is shown in Figure 2.

The movement of the elevation axis is governed by the following differential equations:

𝐽𝜖𝜖̈ = 𝑀𝑐 + 𝑙𝑚𝐹𝑚cos(𝜌) − 𝑀𝑤,𝜖 − 𝑀𝑓,𝜖 (1)

where 𝑀𝑐 is the centrifugal torque, which is function of (𝜖), 𝑀𝑓,𝜖 is friction moment exerted on the helicopter

elevation axis, which is generated from combining joint friction and air resistance, and 𝑀𝑤,𝜖 is the effective

gravitational torque due to the mass differential 𝑊 = 𝑚ℎ𝑔 about the elevation axis.

ISSN: 2088-8708

Int J Elec & Comp Eng, Vol. 10, No. 2, April 2020 : 1868 - 1884

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Figure 1. 3DOF Helicopter System [20]

Figure 2. Schematic diagram of elevation axis model for 3DOF helicopter system

𝑀𝑤,𝜖 = 𝑙𝑚𝑚ℎ𝑔 (2)

If the elevation angle 𝜖 = 0 and the rotation angle of pitch axis 𝜌 = 0, then the centrifugal torque (𝑀𝑐) will

be zero (1) based on Euler’s second law becomes as follows:

𝐽𝜖𝜖̈ = 𝑙𝑚𝐹𝑚 − 𝑀𝑤,𝜖 + 𝑀𝑓,𝜖 (3)

𝐽𝜖𝜖̈ = 𝑙𝑚(𝐹𝑓 + 𝐹𝑏) − 𝑀𝑤,𝜖 + 𝑀𝑓,𝜖 (4)

𝐹𝑖 = 𝐾𝑐𝑉𝑖𝑖 = 𝑓, 𝑏 (5)

𝐽𝜖𝜖̈ = 𝐾𝑐𝑙𝑚(𝑉𝑓 + 𝑉𝑏) − 𝑀𝑤,𝜖 + 𝑀𝑓,𝜖 (6)

𝐽𝜖𝜖̈ = 𝐾𝑐𝑙𝑚𝑉𝑠 − 𝑀𝑤,𝜖 + 𝑀𝑓,𝜖 (7)

2.2.2. Pitch axis model

Consider the pitch schematic diagram of the system in Figure 3. It can be seen from the figure that

the main torque acting on the system pitch axis is produced from the thrust force generated by the properller

motors. When 𝜌 ≠ 0, the gravitational force will also generate a torque 𝑀𝑤,𝜌 acts on the helicopter pitch axis.

𝑀𝑤 ,𝜖 + 𝑀𝑓 ,𝜖

𝜖

𝑙𝑏

𝑙𝑚



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Figure 3. Schematic diagram of pitch axis model for 3DOF helicopter scheme

Table 1. Nomenclature of the helicopter system Nomenclature Definition

𝜖, 𝜌, 𝑟

𝑙𝜌

𝑙𝑚

𝑙𝑏

𝑙ℎ

𝐹𝑚

Rotation angle of elevation, pitch and travel axis respectively (degree)

Distance between the pitch axis and the helicopter center of gravity (m)

Distance from axle point to either propeller motor (m)

Distance from axle point to the balance block (m)

Distance between the pitch axis and the center of propellers (m)

Lift force of the properller motors (N)

𝐹𝑓 , 𝐹𝑏

𝑉𝑓 , 𝑉𝑏

Forces of fornt and back properller motors (N)

Votage applied to the front and back properller motor respectively (v)

𝑚𝑏

𝑚𝑚

𝑊

𝑀𝐹,𝜖 , 𝑀𝐹,𝜌,𝑀𝐹,𝑟

𝑀𝑐

𝑀𝑤,𝜖 ,𝑀𝑤,𝜌

𝑗𝜖 , 𝑗𝜌, 𝑗𝑟

𝐾𝑐

Mass of balance block (kg)

Mass of properller motors (kg)

Mass differential about the elevation axis (kg)

Torque component generated from combining the joint friction and air resistance for elevation, pitch and

rate axis respectively (N.m)

Centrifugal torque about the elevation axis (N.m)

Effective gravitational torque due to gravity about the elevation axis and pitch axis respectively (N.m)

Inertia moment of the system about elevation, pitch and travel axis respectively (kg.m2)

Force constant of the motor-propeller combination

The dynamics of the pitch axis can be modeled mathematically as follows:

𝐽𝜌�̈� = 𝐹𝑓𝑙ℎ − 𝐹𝑏𝑙ℎ − 𝑀𝑤,𝜌 − 𝑀𝑓,𝜌 (8)

Where 𝑀𝑓,𝜌is friction moment exerted on the pitch axis.

𝑀𝑤,𝜌 = 𝑚ℎ𝑔𝑙𝜌 sin(𝜌) cos(𝜖) (9)

Based on the assumption that the pitch angle 𝜌 = 0, 𝑀𝑤,𝜌 = 0 , then (8) becomes as follows:

𝐽𝜌�̈� = 𝑙ℎ(𝐹𝑓 − 𝐹𝑏) − 𝑀𝑓,𝜌 (10)

𝐽𝜌�̈� = 𝐾𝑐𝑙𝜌(𝑉𝑓 − 𝑉𝑏) − 𝑀𝑓,𝜌 (11)

𝐽𝜌�̈� = 𝐾𝑐𝑙𝜌𝑉𝑑 − 𝑀𝑓,𝜌 (12)

2.2.3. Travel axis model The free body diagranm of the helicopter system dynamics based on travel axis is presented in

Figure 4. In this model, when 𝜌 ≠ 0, the main forces acting on the helicopter dynamics are the thrust forces

of properller motors (𝐹𝑓, 𝐹𝑏). These forces have a component generates a torque on the travel axis. Assume

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that the helicopter body has roll up by an angle 𝜌 as shown in Figure 4. Then the dynamics of travel axis for

3DOF helicopter system is modeled as follows:

𝐽𝑟�̇� = −(𝐹𝑓 + 𝐹𝑏)𝑠𝑖𝑛(𝜌)𝑙𝑚 − 𝑀𝑓,𝑟 (13)

Figure 4. Schematic diagram of travel rate axis model for 3DOF helicopter scheme

The thrust foces of the two properller motors (𝐹𝑓 + 𝐹𝑏) are required to keep the helicopter in flight

case and is approximately 𝑊.

𝐽𝑟�̇� = −𝑊𝑠𝑖𝑛(𝜌)𝑙𝑚 − 𝑀𝑓,𝑟 (14)

Where 𝑀𝑓,𝑟 is friction moment exerted on travel axis. As 𝜌 approachs to zero, based on sinc function,

𝑠𝑖𝑛(𝜌) = 𝜌, the above equation becomes as follows:

𝐽𝑟�̇� = −𝑊𝜌𝑙𝑚 − 𝑀𝑓,𝑟 (15)

Based on the assumption that the coupling dynamics, gravitational torque (𝑀𝑤,𝜖) and friction moment exerted

on elevation, pitch and travel axis are neglected, then the dynamics modeling (7), (12) and (15) for 3DOF

helicopter system can be simplified as in (16), (17) and (18) respectively [1].

𝜖̈ =𝐾𝑐𝑙𝑚

𝐽𝜖𝑉𝑠 (16)

�̈� =𝐾𝑐𝑙𝜌

𝐽𝜌𝑉𝑑 (17)

�̇� =𝑊𝑙𝑚

𝐽𝑟𝜌 (18)

2.3. System state space model

In order to design state feedback controller based on LQR technique for 3DOF helicopter system,

the dynamics model of the system should be formulated in state space form. In this study, the proposed

hybrid control algorithm is investigated for the purpose of control of pitch angle, elevation angle and travel

rate of 3DOF helicopter scheme by regulating the voltage suopplies to the front and back motors.

Let 𝑥(nx1) = [𝑥1,𝑥2, 𝑥3,𝑥4, 𝑥5,𝑥6, 𝑥7]𝑇 = [𝜖, 𝜌, 𝜖̇, 𝜌,̇ 𝑟, ʓ, 𝛾]𝑇 be the state vector of the system, the state

variables are chosen as the angles and rate and their corresponding angular velocities, and ʓ̇ = 𝜖, �̇� = 𝑟.

The voltages supplied to the front and back propellers motors are considered the input’s vector such that,

𝑢(𝑡)(mx1) = [𝑢1,𝑢2]𝑇

= [𝑉𝑓, 𝑉𝑏]𝑇 and the elevation angle, pitch angle and travel rate are assumed



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the output’s vector such that, 𝑦(𝑡)(px1) = [𝜖, 𝜌, 𝑟]𝑇. Based on (13)-(15), choosing these state variables

yields the following system state space model:

𝑥1̇ = 𝜌 = 𝑥2

𝑥2̇ = 𝜖̇ = 𝑥3

𝑥3̇ = 𝜖̈ =𝐾𝑐𝑙𝑚𝐽𝜖

(𝑉𝑓 + 𝑉𝑏)

𝑥4̇ = �̈� =𝐾𝑐𝑙𝜌

𝐽𝜌(𝑉𝑓 − 𝑉𝑏) (16)

𝑥5̇ = �̇� =𝑊𝑙𝑚𝐽𝑟

𝑥2

𝑥6̇ = ʓ̇ = 𝑥1

𝑥7̇ = �̇� = 𝑥4

The general state and output matrix equations describing the dynamic behavior of the linear-time-

invariant (LTI) helicopter system in state space form are as follows:

𝑥(̇𝑡) = 𝐴𝑥(𝑡) + 𝐵𝑢(𝑡) (17)

𝑦(𝑡) = 𝐶𝑥(𝑡) + 𝐷𝑢(𝑡) (18)

Where 𝐴(nxn) is the system matrix, 𝐵(nxm) is the input matrix, 𝐶(pxm) is the output matrix, and 𝐷(mxp)

is feed forward matrix, for the designed system. Based on (16), the state space representation (17) and (18)

are rewritten as in (19) and (20) [1].

[ 0 0 1 0 0 0 00 0 0 1 0 0 00 0 0 0 0 0 00 0 0 0 0 0 0

0𝑊𝑙𝑚

𝐽𝑟0 0 0 0 0

1 0 0 0 0 0 00 0 0 0 1 0 0]

[ 𝜖𝜌𝜖̇�̇�𝑟ʓ𝛾]

+

[

0 00 0

𝐾𝑐𝑙𝑚

𝐽𝜖

𝐾𝑐𝑙𝑚

𝐽𝜖𝐾𝑐𝑙𝜌

𝐽𝜌−

𝐾𝑐𝜌

𝐽𝜌

0 00 00 0 ]

[𝑉𝑓

𝑉𝑏] (19)

[𝜖𝜌𝑟] = [

1 0 0 0 0 0 00 1 0 0 0 0 00 0 0 0 1 0 0

]


+ [0 00 00 0

] [𝑉𝑓

𝑉𝑏] (20)

[ 𝜖̇�̇�𝜖̈�̈��̇�ʓ̇�̇�]

=

[ 0 0 1 0 0 0 00 0 0 1 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 2.0655 0 0 0 0 01 0 0 0 0 0 00 0 0 0 1 0 0]


+

[

0 00 0

5.8197 5.819763.9498 −63.9498

0 00 00 0 ]

[𝑉𝑓

𝑉𝑏] (21)

In this study, for purpose of control system design, the model of the system is formulated in state

space form using the physical parameters values stated in Table 2 [1]. Based on (19) and using

the parameters values in Table 1, the state equation of the system is given by (21).

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Table 2. Values of physical parameters of 3DOF helicopter system Symbol Physical Unit Numerical Values

𝐽𝜖 kg.m2 1.8145 𝐽𝑟 kg.m2 1.8145 𝐽𝜌 kg.m2 0.0319 W N 4.2591 𝑙𝑚 m 0.88 𝑙𝑏 m 0.35

𝑙𝜌 m 0.17

𝐾𝑐 N/V 12

3. CONTROLLER TECHNIQUES

In this section, the theory of the PID and LQR controller techniques used for designing the proposed

helicopter control system are introduced. The gain parameters of the PID controller are obtained

approximately from the LQR gain matrix. Control equations of elevation angle, pitch angle, and travel rate

for helicopter system using PID controller are also presented.

3.1. PID controller

A PID is the most popular controller technique that is widely used in industrial applications due to

the simplicity of its structure and can be realized easily for various control problems as the gain parameters of

the controller are relatively independent [21][22]. Basically, the controller provides control command signals

𝑢(𝑡)based on the error 𝑒(𝑡) between the demand input and the actual output of the system. The continuous

time structure of the PID controller is as follows:

𝑢(𝑡) = 𝐾𝑝𝑒(𝑡) +𝐾𝑖 ∫ 𝑒(𝜏)𝑑𝜏𝑡

0+ 𝐾𝑑

𝑑𝑒(𝑡)

𝑑𝑡 (22)

Where 𝐾𝑝, 𝐾𝑖and 𝐾𝑑 are the proportional, integral and differential components of the controller gain.

These controller gain parameters should be tuned properly to enable the output states of the system to

efficiently follow the desired input. If 𝜖𝑑, 𝜌𝑑 and 𝑟𝑑 are the desired elevation angle, pitch angle and travel

rate of the helicopter system, it can express the form of PID controllers used to meet the desired output states

as follows [1,23]. In this study, for elevation angle, the control equation is based on the following PID control

equation:

𝑉𝑠 = 𝐾𝜖𝑝𝑒𝜖 + 𝐾𝜖𝑑𝑒�̇� + 𝐾𝜖𝑖 ∫ 𝑒𝜖𝑑𝑡 (23a)

𝑉𝑠 = 𝐾𝜖𝑝(𝜖𝑑 − 𝜖) − 𝐾𝜖𝑑𝜖̇ + 𝐾𝜖𝑖 ∫(𝜖𝑑 − 𝜖)𝑑𝑡 (23b)

While the pitch angle is controlled by the following PD control equation:

𝑉𝑑 = 𝐾𝜌𝑝𝑒𝜌 + 𝐾𝜌𝑑𝑒�̇� (24a)

𝑉𝑑 = 𝐾𝜌𝑝(𝜌𝑑 − 𝜌) − 𝐾𝜌𝑑�̇� (24b)

The travel rate is governed by the following PI control equation:

𝜌𝑑 = 𝐾𝑟𝑝𝑒𝑟 + 𝐾𝑟𝑖 ∫ 𝑒𝑟𝑑𝑡 (25a)

𝜌𝑑 = 𝐾𝑟𝑝(𝑟𝑑 − 𝑟) + 𝐾𝑟𝑖 ∫(𝑟𝑑 − 𝑟)𝑑𝑡 (25b)

where 𝑒𝜖 = 𝜖𝑑 − 𝜖, 𝑒𝜌 = 𝜌𝑑 − 𝜌, 𝑒𝑟 = 𝑟𝑑 − 𝑟, 𝑒�̇� = −𝜖̇ and 𝑒�̇� = −�̇�.

3.2. LQR controller

Linear quadratic regulator is a common optimal control technique which has been widely utilized in

various manipulator systems [24]. Using LQR technique in design control system requires all the states of

the system plant to be measurable as it bases on the full state feedback concept. Therefore, using LQR

controller to stabilize the 3DOF helicopter system based on the assumption that the states of the system are

considered measurable. LQR approach includes applying the optimal control effort:



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𝑢(𝑡) = −𝐾𝑥(𝑡) (26)

Where 𝐾 is the state feedback gain matrix of LQR controller, that will enable the output states of the system

to follow the trajectories of reference input, while minimizing the following the cost function:

𝐽 = ∫ (𝑥𝑇(𝑡)𝑄𝑥(𝑡)∞

0− 𝑢𝑇(𝑡)𝑅𝑢(𝑡))𝑑𝑡 (27)

Where Q and R are referred to as weighting state and control matrices. The controller feedback gain matrix

can be determined by using below equation:

𝐾 = 𝑅−1𝐵𝑇𝑃 (28)

Where 𝑃 is (nxn) matrix determined from the solution of the following Riccati matrix equation:

𝐴𝑇𝑃 + 𝑃𝐴 − 𝑃𝐵𝑅−1𝐵𝑇𝑃 + 𝑄 = 0 (29)

In this application, 𝐾 = [k11 k12 k13 k14 k15 k16 k17

k21 k22 k23 k24 k25 k26 k27] and 𝑢(𝑡) = [

𝑢1

𝑢2]. Based on this,

the control effort of the system stated in (26) can be written as follows:

[𝑢1

𝑢2] = − [

k11 k12 k13 k14 k15 k16 k17

k21 k22 k23 k24 k25 k26 k27]


(30)

For the purpose of simplicity of control problem the weighting matrices Q and R are chosen as the diagonal

matrices so that the cost function (27) can be reformulated as below:

𝐽 = ∫ (𝑞11𝑥12 + 𝑞22𝑥2

2 + 𝑞33𝑥32 + 𝑞44𝑥4

2 + 𝑞55𝑥52 + 𝑞66𝑥6

2 + 𝑞77𝑥72 + 𝑟11𝑢1

2 + 𝑟22𝑢22)

∞

0𝑑𝑡 (31)

Where 𝑞11, 𝑞22, 𝑞33, 𝑞44, 𝑞55, 𝑞66 and 𝑞77 denote the weighting elements of flight angles and their

corresponding angular velocities of the proposed 3DOF helicopter system respectively, while, 𝑟11and 𝑟22 are

the weighting elements of control inputs. It is worth considering that, in this study, based on the state and

control LQR weighting matrices Q and R, the feedback gain matrix (𝐾) can be calculated by utilizing

the Matlab command “lqr”.

The optimal control approach LQR is highly recommended for stabilizing the 3DOF helicopter

system as it basically looks for a compromising between the best control performance and minimum control

effort. Based on LQR controller, an optimum tracking performance can be investigated by a proper setting of

the feedback controller gain matrix. To achieve this, the LQR controller is optimized by using GA tuning

method which is adopted to obtain an optimum elements values for Q and R weighting matrices.

4. TUNING METHOD

In this study, GA tuning approach is adopted to optimize the LQR gain matrix used to approximate

the gain parameters of PID controller for 3DOF helicopter system. GA is a global search optimization

technique bases on the strategy of genetics and natural selection [25][26]. This optimization method is

utilized to obtain an optimum global solution for more control and manipulating problems. The procedure of

GA approach includes of three basic stepes: Selection, Crossover and Mutation. Applying these main

operations creates new individuals which could be better than their parents. Based on the requirements of

desired response, the sequence of GA optimization technique is repeated for many iterations and finally stops

at generating optimum solution elements for the application problems [27]. The sequence of the GA tuning

method is presented in Figure 5 [3]. The steps of the GA loop are defined as follows:

1. Initial set of population.

2. Selecting individuals for mating.

3. Mating the population to create progeny.

4. Mutate progeny.

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5. Inserting new generated individuals into population.

6. Are the system fitness function satisfied?

7. End search process for solution.

In this study, the aim of using GA optimization method is to tune the PID controller parameters by

minimizing a selected fitness function which will be discussed in detail in the section 4. The implementation

procedure of the GA tuning method begins with the definition step of the chromosome representation. Each

chromosome is represented by nine cells which correspond to the weight matrices elements of the LQR

controller as shown in Figure 6. By this representation it can adjust the LQR elements in order to achieve

the required performance. These cells are formed by real positive numbers and characterize the individual to

be evaluated [27].

Figure 5. Process loop of GA.

Figure 6. Definition of GA chromosome

5. HELICOPTER HYBRID CONTROLS YSTEM DESIGN

In this study, a control system using LQR based PID controller is designed to stabilize 3DOF

helicopter system. Based on step input, a hybrid controller is designed for the following desired performance

parameters: rise time (𝑡𝑟) less than 10 (ms), settling time (𝑡𝑠) less than 30 (ms), maximum overshoot

percentage, (𝑀𝑂) less than 5%. The block diagram of the proposed helicopter control system based on LQR

controller is presented in Figure 7.

Figure 7. LQR controller based on GA for 3DOF helicopter system

11q 22q 33q 44q 55q 66q 77q 11r 22r



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The control system is analysed mathematically and then simulated using Matlab/simulink tool to

validate the proposed LQR controller. Based on the desired performance parameters which include rise and

settling time, overshoot and error steady state parameters the fitness function of the control problem is

formulated as follows:

𝐹 = 0.3𝑆. 𝑡𝑟 + 0.3𝑆. 𝑡𝑠 + 0.2𝑆. 𝑂 + 0.2𝑆. 𝑒𝑠𝑠 (32)

Where, 𝑆 is closed loop transfer function of the helicopter system, 𝑆. 𝑡𝑟 , 𝑡𝑠, 𝑂, 𝑒𝑠𝑠 are the rise time, settling

time, maximum over shoot and error steady state of the closed-loop control system. It is worth considering

that the control input effort is considered in the evaluation process of the proposed stabilizing helicopter

system. In this study, the design of controller is effectively optimized by using GA tuning method which is

adopted to obtain optimum elements values for LQR weighting matrices Q and R. These optimized matrices

are using to calcutate optimum controller gain matrix by using (29) and (30). However, the gain matrix is

determined by using the Matlab command ‘lqr’.

5.1. PID approximation

In this subsection, the gain parameters 𝐾𝑝, 𝐾𝑖 , 𝐾𝑑of the PID controller are calculated approximately

from the feedback gain matrix 𝐾 of the LQR controller based on GA optimization technique. Analyzing (30)

yields the following [4]:

[𝑢1

𝑢2] = − [

k11 k12 k13 k14 k15 k16 k17

k11 −k12 k13 −k14 −k15 k16 −k17]


(33)

5.1.1. Elevation control using PID controller Summing the rows of (33) results the following [1]:

𝑢1 + 𝑢2 = −(2𝑘11𝜖 + 2𝑘13𝜖̇ + 2𝑘16ʓ) = −(2𝑘11𝜖 + 2𝑘13𝜖̇ + 2𝑘16 ∫ 𝜖 𝑑𝑡) (34)

The above equation can be written as

𝑉𝑠 = −2𝑘11(𝜖𝑑 − 𝜖) − 2𝑘13𝜖̇ − 2𝑘16 ∫(𝜖𝑑 − 𝜖) 𝑑𝑡 (35)

It is obvious that (23b) and (35) have the same structure, this means that the gain parameters of pitch PID

controller can be obtained from the gain elements of LQR controller. Thus, comparing (23b) with (35), yields

the following gain relationships:

𝐾𝜖𝑝 = 2𝑘11

𝐾𝜖𝑑 = 2𝑘13

𝐾𝜖𝑖 = 2𝑘16

(36)

The block diagram of closed-loop control system for 3DOF helicopter system based on PID controller is

shown in Figure 8. Taking Laplace transform for elevation axis model (13) yields the following equation:

𝐽𝑒𝜖(𝑠). 𝑠2 = 𝐾𝑐𝑙1𝑉𝑠(𝑠) (37)

The transfer function of the elevation axis plant is given by:

𝜖(𝑠)

𝑉𝑠(𝑠)=

𝐾𝑐𝑙𝑚

𝐽𝜖𝑠2 (38)

The transfer function of the PID controller is as follows:

𝑉𝑠(𝑠)

𝐸𝜖(𝑠)=

𝐾𝜖𝑑𝑠2+𝐾𝜖𝑝𝑠+𝐾𝜖𝑖

𝑠 (39)

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where 𝐸𝜖(𝑠) = 𝜖𝑑(𝑠) − 𝜖(𝑠), the open loop transfer function of the elevation axis control 𝐺𝜖(𝑠) is given by:

𝐺𝜖(𝑠) =𝜖(𝑠)

𝐸𝜖(𝑠)=

𝑉𝑠(𝑠)

𝐸𝜖(𝑠)

𝜖(𝑠)

𝑉𝑠(𝑠) (40)

Based on (38) and (39), the open loop elevation transfer function becomes:

𝐺𝜖(𝑠) =𝐾𝑐𝑙𝑚

𝐽𝜖𝑠2 𝐾𝜖𝑑𝑠2+𝐾𝜖𝑝𝑠+𝐾𝜖𝑖

𝑠 (41)

The closed loop transfer function for elevation angle control is as follows:

𝜖(𝑠)

𝜖𝑐(𝑠)=

𝐺𝜖(𝑠)

1+𝐺𝜖(𝑠)=

𝐾𝑐𝑙𝑚𝐾𝜖𝑑𝑠2+𝐾𝑐𝑙𝑚𝐾𝜖𝑝𝑠+𝐾𝑐𝑙𝑚𝐾𝜖𝑖

𝐽𝑒𝑠3+𝐾𝑐𝑙𝑚𝐾𝜖𝑑𝑠2+𝐾𝑐𝑙𝑚𝐾𝜖𝑝𝑠+𝐾𝑐𝑙𝑚𝐾𝜖𝑖 (42)

Figure 8. Control system block diagram for helicopter elevation, pitch and travel axis using PID controller

5.1.2. Pitch control using PD controller

Similarly, the difference of the rows of (33) results in

𝑢1 − 𝑢2 = −2𝑘12𝜌 − 2𝑘14�̇� − 2𝑘15(𝑟 − 𝑟𝑐) − 2𝑘17𝛾 (43)

𝑉𝑑 = −2𝑘12𝜌 − 2𝑘14�̇� − 2𝑘15(𝑟 − 𝑟𝑑) − 2𝑘17 ∫(𝑟 − 𝑟𝑑)𝑑𝑡 (44)

Substitution (25b) in (24b) results,

𝑉𝑑 = −𝐾𝜌𝑝𝜌 − 𝐾𝜌𝑑�̇� + 𝐾𝜌𝑝𝐾𝑟𝑝(𝑟𝑑 − 𝑟) + 𝐾𝜌𝑝𝐾𝑟𝑖 ∫(𝑟𝑑 − 𝑟)𝑑𝑡 (45)

It is clear that (44) and (45) have exactly the same structure. Then, by comparing these equations, it can

obtain the feedback gains parameters for the PID controller from the LQR gains parameters as follows:

𝐾𝜌𝑝 = 2𝑘12

𝐾𝜌𝑑 = 2𝑘14

𝐾𝑟𝑝 =𝑘15

𝑘12

𝐾𝑟𝑖 =𝑘17

𝑘12

(46)

Taking Laplace transform for pitch axis model (14) yields:

𝐽𝜌𝜌(𝑠)𝑠2 = 𝐾𝑐𝑙𝜌𝑉𝑑(𝑠) (47)

The transfer function for pitch axis model is given by:

ρ(s)

Vd(s)=

𝐾𝑐𝑙𝜌

𝐽𝜌s2 (48)



1879

The transfer function of the PD controller is as follows:

𝑉𝑑(𝑠)

𝐸𝜌(𝑠)= 𝐾𝜌𝑑𝑠 + 𝐾𝜌𝑝 (49)

where 𝐸𝜌(𝑠) = 𝜌𝑑(𝑠) − 𝜌(𝑠), the open loop transfer function of the pitch axis control is given by:

𝐺𝑝(𝑠) =𝜌(𝑠)

𝐸𝜌(𝑠)=

𝜌(𝑠)

𝑉𝑑(𝑠)

𝑉𝑑(𝑠)

𝐸𝜌(𝑠)=

𝐾𝑐𝑙𝜌(𝐾𝜌𝑑𝑠+𝐾𝜌𝑝)

𝐽𝜌𝑠2 (50)

The closed loop transfer function of pitch angle is given by:

𝜌(𝑠)

𝜌𝑐(𝑠)= −

𝐾𝑐𝑙𝜌𝐾𝜌𝑑𝑠+𝐾𝑐𝑙𝜌𝐾𝜌𝑝

𝐽𝜌𝑠2+𝐾𝑐𝑙𝜌𝐾𝜌𝑑𝑠+𝐾𝑐𝑙𝜌𝐾𝜌𝑝 (51)

5.1.3. Travel control using PI controller

Taking Laplace transform for travel axis model (15) results:

𝑟(𝑠)𝑠 =𝑊𝑙𝑚

𝐽𝑟𝜌(𝑠) (52)

The transfer function for travel axis model is given by:

r(s)

ρ(s)=

𝑊𝑙𝑚

𝐽𝑟s (53)

The transfer function of the PI controller is as follows:

𝜌(𝑠)

𝐸𝑟(𝑠)= 𝐾𝑟𝑝 +

𝐾𝑟𝑖

𝑠 (54)

where 𝐸(𝑠) = 𝜌𝑑(𝑠) − 𝜌(𝑠), the open loop transfer function of the travel axis control is given by:

𝐺𝑟(𝑠) =r(s)

𝜌(s)

𝜌𝜌(𝑠)

𝐸𝑟(𝑠)=

𝑊𝑙𝑚(𝐾𝑟𝑝𝑠+𝐾𝑟𝑖)

𝐽𝑟𝑠2 (55)

The closed loop transfer function for travel angle is as follows:

r(s)

rd(s)=

𝑊𝑙𝑚𝐾𝑟𝑝𝑠+𝑊𝑙𝑚𝐾𝑟𝑖

𝐽𝑟𝑠2+𝑊𝑙𝑚𝐾𝑟𝑝𝑠+𝑊𝑙𝑚𝐾𝑟𝑖 (56)

6. CONTROLLER SIMULATION AND RESULTS

6.1. LQR controller

In order to validate the proposed helicopter stabilizing system, the LQR controller is analysed

mathematically using Matlab environment. Based on objective function (J) and using the Matlab command

“lqr” the elements of the LQR weighting matrices Q, R are tuned using GA optimization method.

The parameters of the GA optimization approach chosen for tuning process of the 3DOF helicopter control

system are listed in Table 3. Converging elements of the LQR weighting matrices Q, R through iteration

based on GA optimization method are presented in Figure 9.

Table 3. Parameters of GA tuning method GA Parameter Value/Method

Population Size 20

Max No. of Generations 100

Selection Method Normalized Geometric Selection

Crossover Method Scattering

Mutation Method Uniform Mutation

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Figure 9. Number of generation of GA-LQR parameters Q and R

Based on the fitness function, the optimal elements values for LQR weighting matrices Q, R

obtained based on the GA tuning approach are given by:

𝑄 =

[ 9.0425 0 0 0 0 0 0

0 0.7840 0 0 0 0 00 0 4.5093 0 0 0 00 0 0 0.2763 0 0 00 0 0 0 0.7077 0 00 0 0 0 0 0.9876 00 0 0 0 0 0 0.8004]

, R = [0.0483 0

0 0.0483]

By using the state matrix (𝐴), input matrix (𝐵), and weighting matrices (𝑄, 𝑅), the optimized feedback gain

matrix 𝐾 and closed-loop rictions of the system are determined using the Matlab command “lqr”.

The calculated gain matrix 𝐾 is given by:

𝐾 = [117.138 65.9975 68.4252 16.9323 50.6954 31.954 28.766117.138 −65.9975 68.4252 −16.9323 −50.6954 31.954 −28.766

]

The poles of the closed-loop control system for helicopter using optimized GA-LQR controller are listed in

Table 4. Based on the feedback gain matrix and using (26), the LQR control effort vector for

the 3DOF helicopter system is as follows:

Table 4. Poles of helicopter control system based on LQR technique Pole Value 𝑃1 -794.7130 𝑃2 -1.3745

𝑃3 -0.3405

𝑃4 -2161.7

𝑃5 -1.3623 + i1.0631

𝑃6 -1.3623 - 1.0631i

𝑃7 -1.1773

[𝑢1

𝑢2] = − [

117.138 65.998 68.425 16.932 50.695 31.954 28.766117.138 −65.998 68.425 −16.932 −50.695 31.954 −28.766

]


(57)



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6.2. PID controller

Based on (36), (46) and (57), the absolute values of PID, PD and PI gain parameters for elevation,

pitch and travel axis model respectively for helicopter system are listed in Table 5 [1]. Using the values in

Table 2 and 5, the closed-loop transfer function of elevation, pitch and travel axis (42), (51) and (56) become

as in (58), (59) and (60) respectively:

Table 5. Values of gain parameters for PID, PD and PI controllers PID Parameter Relationship Absolute Value

𝐾𝜖𝑝 2𝑘11 234.2765

𝐾𝜖𝑑 2𝑘13 136.8504

𝐾𝜖𝑖 2𝑘16 63.9086

𝐾𝜌𝑝 2𝑘12 131.9949

𝐾𝜌𝑑 2𝑘14 33.8645

𝐾𝑟𝑝 𝑘15/𝑘12 0.7681

𝐾𝑟𝑖 𝑘17/𝑘12 0.4359

𝜖(𝑠)

𝜖𝑐(𝑠)=

1445𝑠2+247𝑠+674.9

1.8145𝑠3+1445𝑠2+2474𝑠+674.9 (58)

𝜌(𝑠)

𝜌𝑐(𝑠)=

69.08𝑠+269.3

0.0319𝑠2+69.08𝑠+269.3 (59)

𝑟(𝑠)

𝑟𝑐(𝑠)=

2.878𝑠+1.634

1.815𝑠2+2.878𝑠+1.634 (60)

Based on bounded input signal, the elevation, pitch and travel axis model of 3DOF helicopter

system are unstable as they give unbounded outputs. The output time responses for elevation, pitch and travel

angle are illustrated in the Figure 10. In this study, in order to to achieve a stable output, a hybrid control

system using LQR based PID controller for 3DOF helicopter system is proposed to control the dynamic

behavior of the system. To validate the proposed helicopter stabilizing scheme, the controller is simulated

using Matlab programming tool. Three axis, elevation, pitch, travel rate, are considered in the simulation

process of the control system. The performance of the helicopter balancing system is evaluated under unit

step reference input using rise, settling time, overshoot and steady state error parameters for the elevation, pitch and travel angles to simulate the desired command given by the pilot.

6.2.1. Elevation axis model simulation

This section deals with the simulatin of LQR based PID controller used to control the position of

helicopter elevation model. Figure 11 presents tracking control curve of the demand input based on PID

controller for helicopter elevation angle. The simulation result shows that the controller succeeded to guide

the output state of the system through the desired input trajectory effectively with negligible overshoot, short

rise and settling time of 0.1 ms and 0.3 ms respectively.

6.2.2. Pitch axis model simulation

In this section, GA-LQR based PD controller is designed to control the dynamic model of helicopter

pitch angle. Based on the optimized PD gain parameters stated in Table 4, the output response of proposed

helicopter tracking system is illustrated in Figure 12. It is obvious that the controller forced the pitch angle

state to follow the desired trajectory effectively.

6.2.3. Travel axis model simulation

The control of travel rate for 3DOF helicopter system is governed GA-LQR based PI controller.

The output response of the PI tracking scheme using optimum gain parameters listed in Table 5 is shown in

Figure 13. It can be noted that the optimized hybrid controller enabled the output state to track the desired

input trajectory without overshoot, and shorter rise and settling time with minimal steady state tracking error.

Regarding the control effort, the curves of input signals supplied to the propeller motors for the proposed

3DOF helicopter system are shown in Figure 14. It can be seen from input response that the control inputs of

the helicopter control system were within acceptable values. Based on the Figures 11-13, it can say that

the control performance of optimized GA-LQR based PID, PD and PI controllers for helicopter elevation,

pitch and travel axis model respectively was acceptable through tracking the system output states for

the reference input efficiently. From the mini plots of Figures 11-13, the performance parameters of PID,

PD and PI controller for helicopter elevation, pitch and travel axis are listed in the Table 6. The results in

the table show the effectiveness of the optimized hybrid controllers for helicopter system application.

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Figure 10. Open-loop response of helicopter system

Figure 11. Output response of closed-loop helicopter

control system for elevation angle state


control system for pitch angle state


control system for travel rate state

Figure 14. Conrol input of 3DOF helicopter control

system

Table 6. Performance parameters of helicopter controller system Controller Tr (sec) Ts (sec) Mp %

Elevation PID 0.343 0.535 1.1

Pitch PD 0.582 1.05 0

Travel PI 1.17 12.4 5.29

0 5 10 15 200

2000

4000

6000

8000

10000

12000

14000

Time (s)

An

gle

Elevation angle

Pitch angle

Travel angle

0 0.1 0.2 0.30

0.5

1

0 5 10 15 200

0.2

0.4

0.6

0.8

1

Time (s)

Ele

va

tio

n a

ng

le

0 0.02 0.040

0.5

1

0 5 10 15 200

0.2

0.4

0.6

0.8

1

Time (s)

Pitch

an

gle

0 0.02 0.040

0.5

1

0 2 4 6 8 10-1.5

-1

-0.5

0

0.5

1

Time (s)

Co

ntr

ol e

ffo

rt

u1

u2

0 0.02 0.04-1.5

-1

-0.5

0

0.5

1

0 5 10 15 200

0.2

0.4

0.6

0.8

1

1.2

Tra

ve

l ra

te

Time (s)

0 5 100

0.5

1



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7. CONCLUSION

In this study, an efficient hybrid control system has been developed for 3DOF helicopter system.

The dynamics of elevation, pitch and travel axis for helicopter system is modeled mathematically and then

formulated in state space form to enable utilizing state feedback controller technique. In the proposed

helicopter stabilizing scheme, a combination of a conventional PID control with LQR state feedback

controller is adopted to stabilize the elevation, pitch and travel axis of the helicopter scheme. The gain

parameters of the traditional PID controller are determined from the gain matrix of state feedback LQR

controller.

In this research, the LQR controller is optimized by using GA tuning technique. The GA

optimization method has been adopted to find optimum values for LQR gain matrix elements which are

utilized to find best PID gain parameters. The output response of the optimized helicopter control system has

been evaluated based on rise time, setting time, overshoot and steady state error parameters. The simulation

results have shown the effectiveness of the proposed GA-LQR based PID controller to stabilize the helicopter

system at desired values of elevation and pitch angle and travel parameters.

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